Clamavi De Profundis has a recording of “Far Over Misty Mountains Cold.” They sing Howard Shore’s tune to all 20-odd verses of the original poem(s), including a contrasting melody for Durin’s Song.

Personally, I find Shore’s tune less singable than Maury Laws’s from the 1977 animated version, and it’s less intuitive than folk songs are supposed to be. But the Clamavi De Profundis version is very satisfying. Especially, for some reason,

Under the Mountain dark and tall
The King has come unto his hall.
His foe is dead, the Worm of Dread,
And ever so his foes shall fall.